- Thread starter
- #1

#### Mathelogician

##### Member

- Aug 6, 2012

- 35

My knowledge in the field is so basic. So please answer as clear and simple as possible:

Answering to any one (even not all!) would be helpful.

The following 4 questions are the ones i couldn't solve after trying!

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

1-Prove [0,1] is not isometric to [0,2].

2-For which intervals [a,b] in

**R**is the intersection of [a,b] and

**Q**a clopen (closed and open) subset of the metric space

**Q**?

3-Find a metric space in which the boundary of Mr(p)={xEM :d(x,p)< r} is not equal to the sphere of radius r at p ,{xEM : d(x,p)=r}.

4- For a subspace N of a metric space M,prove that if the openness of S c N is equivalent to openness of S in M, then N is open in M.

============================================================

The next is about my proof of the following problem:

My proof : (I use Euclidean metric on R for simplicity of the proof- it can be done for any other metric on R)Let (An) be a nested decreasing sequence of non-empty closed sets in the metric space M.

If M is complete(every Cauchy is convergent) and diam(An) -> 0 as n -> infinity, show thatis exactly one point( note that diam X= sup{d(x,y): x,y E X} ).S=Intersection of (An)

Lets x,y E S. we must show that d(x,y)=0.

Lets d(x,y)= t > 0. By the red part: there exists a N1 such that for all n>N1 , we have diam(An)<t

Let k > n. So diam(Ak)<t.

In other hand, d(x,y) =< diam(Ak)< t. So d(x,y)< t which contradicts the blue.

Now my question is : What was the usage of supposing Ai's as Closed subspaces of M and M's being complete, in this proof?